The space of all possible musical compositions given the constraints of the standard Western tonal and rhythmic system is uncountably vast. We look at a small subsection of this space: Baroque harmony. The rules governing Baroque harmony have been carefully laid out by musical scholars, and the size of this search space becomes tractable if we are given a melody and a key signature to work with. Music is easily represented in numerical form, and is thus an obvious candidate for computer manipulation. We investigate a system using the genetic algorithm that is capable of generating well constructed Baroque-style harmonies given a user-defined melody